Evolving trajectories of COVID-19 curves in India: Prediction using autoregressive integrated moving average modeling.
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Evolving trajectories of COVID-19 curves in India:
Prediction using autoregressive integrated moving
average modeling.
Sudhir Bhandari
SMS Medical College and Hospitals, Jaipur, Rajasthan, India
Amit Tak ( dramittak@gmail.com )
SMS Medical College and Hospitals, Jaipur, Rajasthan, India https://orcid.org/0000-0003-2509-2311
Jitendra Gupta
SMS Medical College and Hospitals, Jaipur, Rajasthan, India
Bhoopendra Patel
Government Medical College, Barmer, Rajasthan, India
Jyotsna Shukla
SMS Medical College and Hospitals, Jaipur, Rajasthan, India
Ajit Singh Shaktawat
SMS Medical College and Hospitals, Jaipur, Rajasthan, India
Sanjay Singhal
SMS Medical College and Hospitals, Jaipur, Rajasthan, India
Abhishek Saini
SMS Medical College and Hospitals, Jaipur, Rajasthan, India
Shivankan Kakkar
SMS Medical College and Hospitals, Jaipur, Rajasthan, India
Amitabh Dube
SMS Medical College and Hospitals, Jaipur, Rajasthan, India
Sunita Dia
Medstar Washington Hospital Center, Washington DC-20010, USA.
Mahendra Dia
North Carolina State University, Raleigh, NC 27695-7609, USA.
Todd C Wehner
North Carolina State University, Raleigh, NC 27695-7609, USA.
Research Article
Keywords: autoregressive integrated moving average, COVID-19, epidemic curve, forecast, mathematical
modeling, prediction
Page 1/15DOI: https://doi.org/10.21203/rs.3.rs-40385/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License.
Read Full License
Page 2/15Abstract
The forecasting of Coronavirus Disease-19 (COVID-19) dynamics is a centerpiece in evidence based
disease management. Numerous approaches that use mathematical modeling have been used to predict
the outcome of the pandemic, including data driven models, empirical and hybrid models. This study was
aimed at prediction COVID-19 evolution in India using a model based on autoregressive integrated
moving average (ARIMA). Retrieving real time data from the Johns Hopkins dashboard from 11 Mar 2020
to 25 Jun 2020 (N = 107 time points) to t the model. The ARIMA (1,3,2) and ARIMA (3,3,1) model t best
for cumulative cases and deaths respectively with minimum Akaike Informaton Criteria. The prediction of
cumulative cases and deaths for next 10 days from 26 Jun 2020 to 05 Jul 2020 showed a trend toward
continuous increment. The predicted root mean square error (PredRMSE) and base root mean square
error (BaseRMSE) of ARIMA(1,3,2) model was 21137 and 166330 respectively. Similarly, PredRMSE and
BaseRMSE of ARIMA(3,3,1) model was 668.7 and 5431 respectively. We propose that data on COVID-19
be collected continuously, and that forecasting continue in real time. The COVID-19 forecast assist
government in resource optimization and evidence based decision making for a subsequent state of
affairs.
Introduction
Coronavirus Disease–19 (COVID–19) caused by novel Severe Acute Respiratory Syndrome Coronavirus 2
(SARS-CoV–2) has caused a pandemic with global devastation to human life and health. SARS-CoV–2 is
closely related to two bat-derived SARS like coronaviruses, bat-SL-CoVZC45 and bat-SL-CoVZXC21. It is
transmitted by human-to-human transmission via droplets or direct contact, and mean incubation period
is 6.4 days [1]. According to World Health Organization, 10,185,374 con rmed cases and 503,862 deaths
have been recorded by 1 Jul 2020 [2]. Further, India recorded 568,092 con rmed cases and 17400 deaths
by the same time [3]. The novelty and rapid spread of SARS CoV– 2 has challenged medical science
across the disciplines of epidemiology, clinical signs and symptoms, pathophysiology, disease
progression and evolution and management and its preventive protocol. Meanwhile, mathematical
models have been used to predict the course of disease and mortality [4]. Such models have potential in
disease management as well as preventive protocols that are cost effective that can help optimum
allocation of resources to manage the disease.
An econometric model has been proposed in the present study to predict and extrapolate transmission of
COVID–19. The model makes use of autoregressive integrated moving average modeling on the
epidemiological dataset of COVID–19. Our objective is to estimate forecast of COVID–19 cumulative
cases and deaths for India. We propose that data on COVID–19 be collected continuously, and that
forecasting continue in real time in order to assist governments in evidence based decision making.
Methods
Page 3/15A retrospective cohort study was run to forecast coronoavirus–19 disease evolution in Indian
subcontinent. The time series of COVID–19 cumulative cases and deaths from 11 Mar 2020 to 25 Jun
2020 (N = 107 time points) was used in prediction using autoregressive integrated moving average
modeling. A useful ARIMA model depends on the number of sample time points, and a good series would
have more than 50 sample points [5].
Data acquisition:
The Indian data of COVID–19 cumulative cases and deaths from 11 March 2020 to 25 Jun 2020 were
sourced from o cial website of Johns Hopkins University Center for Systems Science and Engineering
(https://systems.jhu.edu/) and the repository (https://github.com/CSSEGISandData/COVID–19) [6].
Excel 2010 was used to build the database.
To validate the model, dataset was distributed into a training set and a validation set. The training
dataset from 11 March 2020 to 04 Jun 2020 (N = 86) was used to t autoregressive integrated moving
average (ARIMA) model and estimation of parameters. The validation dataset from 05 Jun 2020 to 25
Jun 2020 N = 21 was used to validate of the model (Table 1).
Model description
The autoregressive integrated moving average model was proposed to estimate the forecast of COVID–
19 evolution [7][8][9]. We followed Box-Jenkins methodology (Figure 1). The methodology encompasses
three phases—identi cation, estimation and testing and, and application. The Identi cation phase
involves data preparation and model selection. The data were analyzed for trends and seasonal
components. The Augmented Dickey-Fuller (ADF) unit root test was performed on time series, to examine
stationarity. Logarithmic transformation and differencing operations were performed to stabilize the time
series. Selection of ARIMA model, was required to establish the order of the autoregressive (AR) process,
‘p’, the differencing operator, ‘d’, and the order of moving average (MA) process, ‘q’. The succeeding
system of mathematical equations delineates the ARIMA (p,d,q) model:
where, {Xt} is the original time series, Wt is the time series acquired after differencing operation on {Xt}, ∇d
is the difference operator,αp,βqare parameters and at is the white noise. Since,
∇d≡(1-B)d
where B is the backward shift operator4, a series {Xt} is integrated of order d if
Page 4/15Wt = (1-B)dXt
The autocorrelation function (ACF) and partial autocorrelation function (PACF) were used to guide the
autoregressive and moving average order respectively. The Akaike Information Criterion (AIC) was used to
select the best model. The AIC contemplate the maximum log likelihood estimation (Log L) and number
of parameters, as the criterion for best model selection. The minimization of the following equation is
required:
AIC = –2logmaximum likelihood+2k
where, k = p+q+1, 2k serves as a penalty function. The model having minimum AIC value was considered
the best. Estimation of parameters was accomplished using the method of maximum likelihood
estimation. The assumptions of the model were scrutinized with residual diagnostics. The forecast of
cumulative cases and deaths were performed from the best selected ARIMA model. The validation of the
model was performed from validation dataset by enumerating Predicted Root Mean Square Error
(PredRMSE) and comparing it with Base Root Mean Square Error (Base RMSE). The forecast of
cumulative cases and deaths was performed from 26 Jun to 05 Jul 2020.
Statistical analysis
The estimates from the ARIMA model follows a normal distribution, and variation of the forecast was
considered within 95% con dence intervals. The MS Excel 2010 was used to maintain the database and
MATLAB 2016a (version 9.0.0.341360) was used for analysis [10].
Results
In order to achieve stationarity, the logarithmic transformation of datasets (cumulative cases and deaths)
were performed as they were showing upward trends. Further difference operations were performed on
cumulative cases (d = 3) and death (d = 2) datasets. The Augmented Dickey-Fulller (ADF) test showed
stationarity (P value < 0.05). The autocorrelation function and partial autocorrelation functions were
calculated and correlograms were used to guide the autoregressive order and the moving average order
of the ARIMA models (Supplemental Figure 1 and 2). The ARIMA (1,3,2) and ARIMA (3,3,1) model for
cumulative cases (AIC = 2.32) and death cases (AIC = 108.75) was chosen respectively with lowest AIC
values. Goodness of t and model assumptions were tested with residual analysis. The graph of
standardized residuals showed random distribution of residuals. The middle values of the QQ plot were
on a straight line. The autocorrelation function and partial autocorrelation functions within random limits,
showed normal distribution and independence (Supplemental Figure 3 and 4). The parameters of both
the ARIMA models were estimated (Supplemental Table S1 and S2).
Model validation was conducted by detecting the differences between the observed values and predicted
values from the validation dataset. The PredRMSE and BaseRMSE of ARIMA(1,3,2) model for time series
Page 5/15of cumulative cases was 21137 and 166330 respectively. Similarly, PredRMSE and BaseRMSE of
ARIMA(3,3,1) model for time series of death cases was 668.7 and 5431 respectively. Lower values of
PredRMSE than Base RMSE indicated a good t (Table 2).The estimation of forecast for cumulative
cases and death cases were performed at 95% con dence interval (Figure 2 and 3; Table 3).
Discussion
The dynamics of any infectious disease involves interactions of three elements - host, agent and
environment [11]. India lies in both the Northern and Eastern hemispheres, with latitudes 8°4' to 37°6'
north and longitudes 68°7' to 97°25' east results in high environmental variability, and subsequent effects
on human behavior and society [12]. Mathematical models are a centerpiece to forecasting the dynamics
of infectious disease pandemics. The fundamental models used in prediction include data driven ( [13]
[14][15][16][17]) empirical [18][19][20], hybrid [21][22] models.The parameters of empirical models
(incubation period, attack rate and recovery rate) possess probability distributions such as Ehrlang and
the Poisson distribution [11]. The data driven models include ARIMA, single-input and single-output
(SISO) model [23] and AI based models (machine learning and deep learning techniques) [14]. The
signi cance of ARIMA model lies in modeling non-stationary time series. The ARIMA model assumes the
trends will continue in the future inde nitely as against the empirical models which assumes
convergence. Few studies used non-parametric models like fourier decomposition methods to predict turn
around dates of the epidemic and the results were found to agree with popular SIR models [24].
Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV–2) expressed its presence in India with the
rst case diagnosed on 30 Jan 2020. On 01 Jul 2020 the cumulative cases and deaths in India reached
568,092 and 17400 respectively [3]. India ranked 57th among the 100 countries in Global Health Security
Index 2019, a scale to gauge preparedness for the outbreak of serious infectious diseases [25].The
forecast of COVID–19 cases helps government agencies in early preparedness to combat subsequent
state of affairs. The patients of COVID–19 can be classi ed into con rmed cases, recovered cases,
admitted and death cases. All categories have differential importance from the management point of
view. The requirement of hospital beds, medical equipment, hospital staff is a function of number of
admitted cases. Similarly, procurement of plasma for antibody therapy is a function of number of
recovered cases.
The present study forecast the cumulative cases and deaths for India from 26 Jun 2020 to 05 Jul 2020.
The forecast shows continuously increasing trends for both cumulative cases and deaths and shows no
decreasing trends till October 2020.
The ARIMA model was proposed by various authors for forecasting the COVID–19 evolution. The data
from 10 Jan 2020 to 20 Feb 2020 was used to t ARIMA (1,0,4) and ARIMA (1,0,3) model for cumulative
diagnoses and newer diagnoses to forecast next two days [26]. The ARIMA and wavelet hybrid model
was proposed to forecast ten time points of cumulative cases from 05 Apr 2020 to 14 Apr 2020 for India.
The forecast showed oscillations, may be due to effects of lockdown [22]. In another data driven model,
Page 6/15using bi-directional LSTM (long short term memory) model, 15 days prediction of actual cases in India
from 30 Apr 2020 to 14 May 2020 showed error of less than 3% [21]. Susceptible-Exposed-Infectious-
Recovered (SEIR) model was used to predict cumulative cases of COVID–19 in India during lockdown
and post lockdown. The model predicts peak of cumulative cases around 43,000 in mid-May. However,
7–21% increase in peak value of cases was predicted for post lockdown period, re ecting relaxation in
control strategies [20]. The evolution of COVID–19 in topmost affected states of India was done using
SISO model [23]. The most severely affected states were Maharashtra, Gujarat, Tamil Nadu, Delhi and
Rajasthan [23][27][28][29]. One time series analysis used genetic programming to predict the COVID–19
evolution in India. On 13 May 2020, the cumulative cases and death cases were 80,000 and 2500
respectively. The prediction for next ten days was done with 142,000 and 4,200 cumulative cases and
deaths respectively on 23 May 2020 [30]. Sujath used linear regression, multilayer perceptron and vector
autoregressive method and 80 time points till 10 Apr 2020, to predict con rmed cases, deaths and
recovered cases from 11 Apr 2020 to 18 Jun 2020. Although prediction varies across the methods and
did not seems very accurate[31]. Yadav used six regression analysis based machine learning models for
prediction and found six degree polynomial model predict very close to observed data [32].
As the epidemic continues, the effects of different interventional strategies inherited in the time series,
and thus data driven models seems to be more accurate than empirical models. The real-time data
modeling has been proposed for monitoring trends of the Indian subcontinent. Evidence based
interventions should be implemented to control the pandemic.
Conclusion
The present study produces encouraging results with the potential to serve as a good adjuvant to existing
models for continuous predictive monitoring of the COVID–19 pandemic. The forecast of COVID–19 may
assist public health authorities and governmental agencies for early preparedness and evidence based
decision making.
Limitations Of The Study
Firstly, ARIMA model used in forecasting does not capture the non-linear and chaotic dynamics of the
pandemic. Secondly, parameter selection procedure requires repetition with each time series update.
Declarations
Contribution of Authors: All authors contributed equally.
Competing Interest: The authors declare that they have no known competing nancial interests or
personal relationships that could have appeared to in uence the work reported in this paper
Data Availability— Data is available on reasonable request from corresponding author.
Page 7/15Ethics Declarations: The present study used de-identi ed data from public domain, thus exempted from
ethical review by Institutional Ethics Committee.
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Tables
Table 1. Distribution of complete datasets of cumulative cases and death cases into training and
validation subsets.
Page 10/15Sub-datasets Cumulative cases Death cases
Training set 11 Mar - 04 Jun 2020 11 Mar - 04 Jun 2020
(N=86) (N=86)
Validation set 05 Jun - 25 Jun 2020 05 Jun - 25 Jun 2020
(N= 21) (N= 21)
Table 2. Goodness of tted models for cumulative cases and deaths cases using prediction root mean
square error and base root mean square error.
Dataset Model PredRMSE BaseRMSE
Cumulative cases ARIMA (1,3,2) 21137 166330
Death cases ARIMA (3,3,1) 668.70 5431
*ARIMA: autoregressive integrated moving average; PredRMSE: predicted root mean square error;
BaseRMSE: base root mean square error.
Table 3. Forecast of cumulative cases and death cases using ARIMA(1,3,2) and ARIMA (3,3,1) models
respectively. The dataset used for prediction is from 11 Mar 2020 to 25 Jun 2020 (N= 107 time points).
Page 11/15Cumulative cases Cumulative deaths
Date Forecast data 95% CI Forecast 95% CI
LL UL LL UL
26 Jun 20 508210 378270 638150 15732 12399 19065
27 Jun 20 526760 396820 656700 16243 12910 19575
28 Jun 20 546010 416070 675950 16711 13379 20044
29 Jun 20 566050 436100 695990 17198 13865 20530
30 Jun 20 586900 456960 716840 17740 14408 21073
1 Jul 20 608630 478690 738570 18314 14982 21647
2 Jul 20 631290 501350 761230 18873 15540 22205
3 Jul 20 654940 525000 784890 19480 16147 22813
4 Jul 20 679660 549710 809600 20141 16808 23474
5 Jul 20 705500 575560 835440 20818 17485 24150
*CI: con dence interval; LL: lower limit of CI; UL: upper limit of CI.
Figures
Page 12/15Figure 1
Flowchart outline Box Jenkins Methodology for implantation of autoregressive integrated moving
average modeling
Page 13/15Figure 2
Line plot shows observed (black line plot) and predicted (magenta line plot) cumulative cases of COVID-
19 with 95% con dence interval (green line plot). The forecast is based on ARIMA (1,3,2) model tted with
time series of cumulative cases from 11 March to 25 Jun 2020.
Page 14/15Figure 3
Line plot shows observed (black line plot) and predicted (magenta line plot) death cases of COVID-19
with 95% con dence interval (green line plot). The forecast is based on ARIMA (3,3,1) model tted with
time series of death cases from 11 March to 25 Jun 2020.
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